Wow -
Let See - in NY (Corning is on the Southern NY Border with PA)
There is the Finger Lakes Trail (FLT) - Both the Main trunk and the Bristol Hills Branch (BHB)- There are a multitude of places with streams and falls. You can usually get the page maps for the area near you at EMS.
In PA there is within a 2 hour Drive:
The West Rim Trail (WRT) (Pine Creek - PA's Grand Canyon)
The WRT is high above on the rim of the canon for the most part.
The Pine Creek bike path is down below, in the canyon and runs for at least 40 miles with very few road crossings (none for the first 20 some miles) with several areas to camp along Pine Creek
The Black Forest Trail (BFT) - Beautiful Vistas and very secluded primitive campsites along stream junctions.
The Susquehannock Trail (STS)- Follows old logging and tote roads as well as old Shay locomotive Rail paths - camping next to secluded streams.
Many more roads crossings and loops with other trails make the Black forest and the STS great and easy 2-3 day loop trails.
Old Loggers Path (OLP)- Very secluded 23 mile loop (or two smaller loops via road crossings) make this a great 2-3 day circuit hike as well.
These are all easily within 2 hours of Corning. Our cottage is in Hammondsport - about 20 mins from Corning, so I do get to spend a lot of time up there, unfortunately with 2 young kids, I get very little outdoor time lately.
OK Info - Check with Chuck Dillon at Pine Creek Outfitters in Ansonia (Rte 6 PA) He has written several trail guides for many trails up there and shuttle folks to trail-heads.
Also, I have taken my Brittany with me on the BFT, OLP and STS with no issues. There is little foot traffic save the occasional 10-12 person college course group and a few day hikers.
The trails in PA are fantastic, well-marked and for the most part nicely graded (some steeper spots on the BFT, but nothing like the Whites or Daks).
All campsites are unmarked primitive tent sites along streams or on vistas.
My favorites for seclusion, vistas and streams in this order: BFT, STS, OLP, WRT.
PS I like PA trails much, much better than the FLT, though the FLT has LT's if a shelter is important to you.